Question

Circles are drawn through the point (2, 0) to cut intercept of length 5 units on the x-axis. If their centres lie in the first quadrant, then their equation is.

• A. $x ^ { 2 } + y ^ { 2 } + 9 x + 2 f y + 14 = 0$
• B. $3 x ^ { 2 } + 3 y ^ { 2 } + 27 x - 2 f y + 42 = 0$
• C. $x ^ { 2 } + y ^ { 2 } - 9 x + 2 f y + 14 = 0$
• D. $x ^ { 2 } + y ^ { 2 } - 2 f y - 9 y + 14 = 0$

Answer 1

Answer: (C)

$x ^ { 2 } + y ^ { 2 } - 9 x + 2 f y + 14 = 0$

Answer 2

The circle g, f, c passes through (2, 0)

$\therefore$ $4 + 4 g + c = 0$ ….(i)

Intercept on x-axis is $2 \sqrt { \left( g ^ { 2 } - c \right) } = 5$

$\therefore 4 \left( g ^ { 2 } + 4 g + 4 \right) = 25$ by (i)

or $( 2 g + 9 ) ( 2 g - 1 ) = 0 \Rightarrow g = - \frac { 9 } { 2 } , \frac { 1 } { 2 }$

Since centre $( - g , - f )$ lies in 1^st quadrant, we choose $g = - \frac { 9 } { 2 }$ so that $- g = \frac { 9 } { 2 }$ (positive).

$\therefore c = 14$ , (from (i)).